生成一个样本
import numpy as np
from scipy import stats
'''
在这里，我们将t分布的所需形状参数（在统计中对应于自由度）设置为10。使用size = 1000表示我们的样本包含1000个独立绘制的（伪）随机数。由于未指定关键字参数loc和scale，因此将它们设置为默认值零和一。
'''
np.random.seed(2020)
x=stats.t.rvs(10,size=1000)
d:\Anaconda3\lib\importlib\_bootstrap.py:219: RuntimeWarning: numpy.ufunc size changed, may indicate binary incompatibility. Expected 192 from C header, got 216 from PyObject
return f(*args, **kwds)
d:\Anaconda3\lib\importlib\_bootstrap.py:219: RuntimeWarning: numpy.ufunc size changed, may indicate binary incompatibility. Expected 192 from C header, got 216 from PyObject
return f(*args, **kwds)
x
array([-1.79937219e+00, -9.16947599e-01, -6.23672658e-02, 6.93167535e-01,
1.80118163e+00, -1.02592060e+00, 4.01066025e-01, -1.91015338e-01,
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9.23176296e-01, -1.57182499e+00, -6.10276616e-01, 6.33556483e-01,
-8.29672373e-01, -9.51414725e-02, -1.01612493e+00, -1.65914088e+00,
1.50240690e-02, -4.04605525e-01, 4.54091957e-01, 1.96104806e-01,
1.92194868e+00, -1.56156827e+00, 1.32702829e+00, 3.66661037e-02,
1.26848221e-01, -1.07561704e-02, 9.33018317e-01, -1.52379256e-01])
import pandas as pd
df_x=pd.DataFrame(x)
df_x.describe().T
| count | mean | std | min | 25% | 50% | 75% | max | |
|---|---|---|---|---|---|---|---|---|
| 0 | 1000.0 | -0.072891 | 1.081381 | -4.516709 | -0.739328 | -0.050502 | 0.563663 | 3.618239 |
描述性统计
x.mean()
-0.07289058666672985
x.min()
-4.5167089414384085
x.max()
3.618238525954051
x.var()
1.168214980664953
m, v, s, k = stats.t.stats(10, moments='mvsk')
m
array(0.)
v
array(1.25)
s
array(0.)
k
array(1.)
stats.t.stats(10, moments='mvsk')
(array(0.), array(1.25), array(0.), array(1.))
sstr = '%-14s mean = %6.4f, variance = %6.4f, skew = %6.4f, kurtosis = %6.4f'
print(sstr % ('distribution:', m, v, s ,k))
distribution: mean = 0.0000, variance = 1.2500, skew = 0.0000, kurtosis = 1.0000
'''
注意：stats.describe使用无偏估计量作为方差，而np.var是有偏估计量。
对于我们的样本，样本统计量与理论值相差很小。
'''
n, (smin, smax), sm, sv, ss, sk = stats.describe(x)
print(sstr % ('sample:', sm, sv, ss, sk))
sample: mean = -0.0729, variance = 1.1694, skew = 0.0162, kurtosis = 0.8654
T检验和KS检验
print('t-statistic = %6.3f pvalue = %6.4f' % stats.ttest_1samp(x, m))
t-statistic = -2.132 pvalue = 0.0333
tt = (sm-m)/np.sqrt(sv/float(n)) # t-statistic for mean
pval = stats.t.sf(np.abs(tt), n-1)*2 # two-sided pvalue = Prob(abs(t)>tt)
print('t-statistic = %6.3f pvalue = %6.4f' % (tt, pval))
t-statistic = -2.132 pvalue = 0.0333
Kolmogorov-Smirnov检验可用于检验样本来自标准t分布的假设
print('KS-statistic D = %6.3f pvalue = %6.4f' % stats.kstest(x, 't', (10,)))
KS-statistic D = 0.057 pvalue = 0.0027
print('KS-statistic D = %6.3f pvalue = %6.4f' % stats.kstest(x, 'norm'))
KS-statistic D = 0.053 pvalue = 0.0069
d, pval = stats.kstest((x-x.mean())/x.std(), 'norm')
print('KS-statistic D = %6.3f pvalue = %6.4f' % (d, pval))
KS-statistic D = 0.039 pvalue = 0.0871
注意：Kolmogorov-Smirnov检验假设我们针对给定参数的分布进行检验，因为在最后一种情况下，我们估计均值和方差，因此违反了该假设，并且检验统计量的分布（p值是根据，是不正确的。
